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I have a non-metric distance $d$ satisfying positivity and symmetry but not triangle inequality and not identity of indiscernibles. I am interested in the topology induced by this distance. What I found so far is:

  • The balls $B(x,\epsilon) = \{y | d(x,y) < \epsilon\}$ do not form a basis. We can find two balls and a point x in the intersection such that there is no ball around x which fits completely into the intersection.

So what now? Standard literature only treats metric spaces. What if we only have a distance-function? Is it possible at all to describe a topology which is 'compatible' with such a non-metric distance?

In some sense I feel that the topology (if exists) is finer than the Euclidean topology. Let $d'$ be the Euclidean metric. We have

$d(x,y) \leq d'(x,y)$ for all x,y.

My idea is to study the space in Euclidean topology, so all topological properties (like continuity of functions) apply also to the distance space. Does this make sense?

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Those balls may not form a basis, as you correctly say. But they do form a subbasis, and you can then certainly study the topology generated by that subbasis. And once you have introduced these kinds of topological spaces, it certainly makes sense to study them by applying all the ordinary tools and concepts from the study of topology, including continuity.

However, the very spare properties that you have imposed on $d$ do not imply that the topology generated by $d$ is finer than the Euclidean topology. For example, you said that $d$ does not need to satisfy the identity of indiscernibles, and so the constant function $d(x,y)=0$ is an example of your kind of distance function; but the only open sets in the topology generated by that example are the empty set and the whole space.

An issue you might ponder is this. It's not very common for a randomly chosen generalization to be fruitful. Most fruitful generalizations arise from some motivation, perhaps from some "naturally" occurring examples and observation of their properties, or some possibility of useful application. One would want to know: What is the motivation for studying these kinds of topological spaces? What interesting examples are there? What interesting applications might arise from studying these spaces?

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  • $\begingroup$ Thanks a lot. I will have a look on subbases. I actually have a concrete distance $d$ based on dynamic time warping (DTW). Current literature studies DTW distance spaces (e.g. $(\mathbb{R}^n,d)$ ) in Euclidean topology, and this is fine for most practical applications. However, I am curious about how to describe the actual topology generated by this distance function, and how things behave in this topology. We have that the Euclidean Ball is always contained in the DTW Ball (for all identical centers and radii). Does this tell us anything? $\endgroup$
    – powermod
    Jan 30, 2020 at 16:06
  • $\begingroup$ More detailed questions with solid motivation like that will generally work well on this site. But I'll say that the information you've given so far does not distinguish your DTW distance from the constant function $d(x,y)=0$ in my answer. $\endgroup$
    – Lee Mosher
    Jan 30, 2020 at 18:20

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